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Lecture Notes Combinatorics Lecture Notes Combinatorics Lecture by Maria Axenovich and Torsten Ueckerdt (KIT) Problem Classes by Jonathan Rollin (KIT) Lecture Notes by Stefan Walzer (TU Ilmenau) Last updated: May 30, 2017 1 Contents 0 What is Combinatorics?4 1 Permutations and Combinations9 1.1 Basic Counting Principles......................9 1.1.1 Addition Principle...................... 10 1.1.2 Multiplication Principle................... 10 1.1.3 Subtraction Principle.................... 11 1.1.4 Bijection Principle...................... 11 1.1.5 Pigeonhole Principle..................... 11 1.1.6 Double counting....................... 11 1.2 Ordered Arrangements { Strings, Maps and Products...... 12 1.2.1 Permutations......................... 13 1.3 Unordered Arrangements { Combinations, Subsets and Multisets................ 15 1.4 Multinomial Coefficients....................... 16 1.5 The Twelvefold Way { Balls in Boxes............... 22 1.5.1 U L: n Unlabeled Balls in k Labeled Boxes...... 22 1.5.2 L ! U: n Labeled Balls in k Unlabeled Boxes...... 25 1.5.3 L ! L: n Labeled Balls in k Labeled Boxes....... 27 1.5.4 U ! U: n Unlabeled Balls in k Unlabeled Boxes.... 28 1.5.5 Summary:! The Twelvefold Way............... 30 1.6 Binomial Coefficients { Examples and Identities.......... 30 1.7 Permutations of Sets......................... 34 1.7.1 Generating Permutations.................. 36 1.7.2 Cycle Decompositions.................... 37 1.7.3 Transpositions........................ 39 1.7.4 Derangements......................... 42 2 Inclusion-Exclusion-Principle and M¨obiusInversion 45 2.1 The Inclusion-Exclusion Principle.................. 45 2.1.1 Applications......................... 47 2.1.2 Stronger Version of PIE................... 52 2.2 M¨obiusInversion Formula...................... 53 3 Generating Functions 58 3.1 Newton's Binomial Theorem..................... 62 3.2 Exponential Generating Functions................. 64 3.3 Recurrence Relations......................... 67 n 3.3.1 Special Solution an = x and the Characteristic Polynomial 70 3.3.2 Advancement Operator................... 76 3.3.3 Non-homogeneous Recurrences............... 80 3.3.4 Solving Recurrences using Generating Functions..... 82 2 4 Partitions 85 4.1 Partitioning [n] { the set on n elements............... 85 4.1.1 Non-Crossing Partitions................... 86 4.2 Partitioning n { the natural number................ 87 4.3 Young Tableau............................ 93 4.3.1 Counting Tableaux...................... 99 4.3.2 Counting Tableaux of the Same Shape........... 100 5 Partially Ordered Sets 107 5.1 Subposets, Extensions and Dimension............... 111 5.2 Capturing Posets between two Lines................ 117 5.3 Sets of Sets and Multisets { Lattices................ 123 5.3.1 Symmetric Chain Partition................. 126 5.4 General Lattices........................... 130 6 Designs 132 6.1 (Non-)Existence of Designs..................... 133 6.2 Construction of Designs....................... 135 6.3 Projective Planes........................... 137 6.4 Steiner Triple Systems........................ 138 6.5 Resolvable Designs.......................... 139 6.6 Latin Squares............................. 140 3 What is Combinatorics? Combinatorics is a young field of mathematics, starting to be an independent branch only in the 20th century. However, combinatorial methods and problems have been around ever since. Many combinatorial problems look entertaining or aesthetically pleasing and indeed one can say that roots of combinatorics lie in mathematical recreations and games. Nonetheless, this field has grown to be of great importance in today's world, not only because of its use for other fields like physical sciences, social sciences, biological sciences, information theory and computer science. Combinatorics is concerned with: Arrangements of elements in a set into patterns satisfying specific rules, • generally referred to as discrete structures. Here \discrete" (as opposed to continuous) typically also means finite, although we will consider some infinite structures as well. The existence, enumeration, analysis and optimization of discrete struc- • tures. Interconnections, generalizations- and specialization-relations between sev- • eral discrete structures. Existence: We want to arrange elements in a set into patterns satisfying certain rules. Is this possible? Under which conditions is it possible? What are necessary, what sufficient conditions? How do we find such an arrangement? Enumeration: Assume certain arrangements are possible. How many such arrangements exist? Can we say \there are at least this many", \at most this many" or \exactly this many"? How do we generate all arrangements efficiently? Classification: Assume there are many arrangements. Do some of these arrangements differ from others in a particular way? Is there a natural partition of all arrangements into specific classes? Meta-Structure: Do the arrangements even carry a natural underlying structure, e.g., some ordering? When are two arrangements closer to each other or more similar than some other pair of arrangements? Are different classes of arrangements in a particular relation? Optimization: Assume some arrangements differ from others according to some measurement. Can we find or characterize the arrangements with maximum or minimum measure, i.e. the \best" or \worst" ar- rangements? 4 Interconnections: Assume a discrete structure has some properties (num- ber of arrangements, . ) that match with another discrete structure. Can we specify a concrete connection between these structures? If this other structure is well-known, can we draw conclusions about our structure at hand? We will give some life to this abstract list of tasks in the context of the following example. Example (Dimer Problem). Consider a generalized chessboard of size m n (m rows and n columns). We want to cover it perfectly with dominoes of size× 2 1 or with generalized dominoes { called polyominoes { of size k 1. That means× we want to put dominoes (or polyominoes) horizontally or vertically× onto the board such that every square of the board is covered and no two dominoes (or polyominoes) overlap. A perfect covering is also called tiling. Consider Figure 1 for an example. ! Figure 1: The 6 8 board can be tiled with 24 dominoes. The 5 5 board cannot be tiled with× dominoes. × Existence If you look at Figure1, you may notice that whenever m and n are both odd (in the Figure they were both 5), then the board has an odd number of squares and a tiling with dominoes is not possible. If, on the other hand, m is even or n is even, a tiling can easily be found. We will generalize this observation for polyominoes: Claim. An m n board can be tiled with polyominoes of size 1 k if and only if k divides m or× n. × Proof. \ " If k divides m, it is easy to construct a tiling: Just cover every column( with m=k vertical polyominoes. Similarly, if k divides n, cover every row using n=k horizontal polyominoes. \ " Assume k divides neither m nor n (but note that k could still divide ) the product m n). We need to show that no tiling is possible. We · write m = s1k + r1, n = s2k + r2 for appropriate s1; s2; r1; r2 N and 0 < r ; r < k. Without loss of generality, assume r r (the argument2 1 2 1 ≤ 2 is similar if r2 < r1). Consider the colouring of the m n board with k colours as shown in Figure2. × 5 1 2 3 4 5 6 7 8 9 1 2 3 4 k 5 k 6 7 8 Figure 2: Our polyominoes have size k 1. We use k colours (1 = white, k = black) to colour the m n board (here:× k = 6, m = 8, n = 9). Cutting the board at coordinates that× are multiples of k divides the board into several chunks. All chunks have the same number of squares of each color, except for the bottom right chunk where there are more squares of color 1 (here: white) than of color 2 (here: light gray). Formally, the colour of the square (i; j) is defined to be ((i j) mod k)+1. Any polyomino of size k 1 that is placed on the board will− cover exactly one square of each colour.× However, there are more squares of colour 1 than of colour 2, which shows that no tiling with k 1 dominoes is possible. × Indeed, for the number of squares coloured with 1 and 2 we have: # squares coloured with 1 = ks1s2 + s1r2 + s2r1 + r2 # squares coloured with 2 = ks s + s r + s r + r 1 1 2 1 2 2 1 2 − Now that the existence of tilings is answered for rectangular boards, we may be inclined to consider other types of boards as well: Claim (Mutilated Chessboard). The n n board with bottom-left and top-right square removed (see Figure3) cannot be× tiled with (regular) dominoes. Figure 3: A \mutilated" 6 6 board. The missing corners have the same colour. × Proof. If n is odd, then the total number of squares is odd and clearly no tiling can exist. If n is even, consider the usual chessboard-colouring: In it, the missing squares are of the same colour, say black. Since there was an equal number of black and white squares in the non-mutilated board, there are now two more white squares than black squares. Since dominoes always cover exactly one black and one white square, no tiling can exist. 6 Other ways of pruning the board have been studied, but we will not consider them here. Enumeration A general formula to determine the number of ways an m n board can be tiled with dominoes is known. For an 2m 2n board the following× formula is due to Temperly and Fisher [TF61] and independently× Kasteleyn [Kas61] m n mn 2 iπ 2 jπ 4 cos 2m+1 + cos 2n+1 : i=1 j=1 Y Y The special case of an 8 8 board is already non-trivial: × Theorem (Fischer 1961). There are 24 172 532 = 12; 988; 816 ways to tile the 8 8 board with dominoes. · · × Classification Consider tilings of the 4 4 board with dominoes. For some of these tilings there is a vertical line through× the board that does not cut through any domino.
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